where \(p_+(x,v,t)\)  and \(p_-(x,v,t)\)  are the distributions for particles subject to a forward and backward force.
The entropy production for this model in the overdamped regime, or \(\tau _r=0\) , has been obtained in [1]}.
For the walls located at \(x=\pm h\)  it reads
\(T\Pi = \langle \dot{q}\rangle = \frac{v_0^2}{\mu }\left[ \frac{D}{v_0^2\tau } \frac{ \cosh (kh) - \frac{\sinh (kh)}{kh} }{ \frac{D}{v_0^2\tau }\cosh (kh) + \frac{\sinh (kh)}{kh} } \right],\) 
